Newton's law of universal gravitation states that the gravitational force of attraction between any two particles of masses
separated by a distance r
has the magnitude
where G is the universal gravitational constant, which has the value
If we have to find the resultant gravitational force acting on the particle m due to a number of masses M1, M2,….Mn etc. we use the principle of superposition. Let F1, F2,….Fn be the individual forces due to M1, M2,….Mn, each given by the law of gravitation. From the principle of superposition each force acts independently and uninfluenced by the other bodies. The resultant force FR is then found by vector addition
FR = F1 + F2 +……+ Fn
Kepler's laws of planetary motion state that
(a) All planets move in elliptical orbits with the Sun at one of the focal points
(b) The radius vector drawn from the sun to a planet sweeps out equal areas in equal time intervals. This follows from the fact that the force of gravitation on the planet is central and hence angular momentum is conserved.
(c) The square of the orbital period of a planet is proportional to the cube of the semi-major axis of the elliptical orbit of the planet
The period T and radius R of the circular orbit of a planet about the Sun are related by
where Ms is the mass of the Sun. Most planets have nearly circular orbits about the Sun. For elliptical orbits, the above equation is valid if R is replaced by the semi-major axis, a.
The acceleration due to gravity at a height h above the Earth's surface
at depth d below the Earth's surface is
The gravitational force is a conservative force, and therefore a potential energy function can be defined. The gravitational potential energy associated with two particles separated by a distance r is given by
where V is taken to be zero at r →∞. The total potential energy for a system of particles is the sum of energies for all pairs of particles, with each pair represented by a term of the form given by above equation. This prescription follows from the principle of superposition.
If an isolated system consists of a particle of mass m moving with a speed v in the vicinity of a massive body of mass M, the total mechanical energy of the particle is given by
That is, the total mechanical energy is the sum of the kinetic and potential energies. The total energy is a constant of motion.
If m moves in a circular orbit of radius a about M, where
, the total energy of the system is
The escape speed from the surface of the Earth is and has a value of 11.2 km s–1.
If a particle is outside a uniform spherical shell or solid sphere with a spherically symmetric internal mass distribution, the sphere attracts the particle as though the mass of the sphere or shell were concentrated at the centre of the sphere.
If a particle is inside a uniform spherical shell, the gravitational force on the particle is zero. If a particle is inside a homogeneous solid sphere, the force on the particle acts toward the centre of the sphere. This force is exerted by the spherical mass interior to the particle.
A geostationary (geosynchronous communication) satellite moves in a circular orbit in the equatorial plane at a approximate distance of km from the Earth's centre.
A 400 kg satellite is in a circular orbit of radius about the Earth. How much energy is required to transfer it to a circular orbit of radius? What are the changes in the kinetic and potential energies?
The kinetic energy is reduced and it mimics ΔE, namely,
The change in potential energy is twice the change in the total energy, namely
The planet Mars has two moons, phobos and delmos. (i) phobos has a period 7 hours, 39 minutes and an orbital radius of . Calculate the mass of mars. (ii) Assume that earth and mars move in circular orbits around the sun, with the martian orbit being 1.52 times the orbital radius of the earth. What is the length of the martian year in days?
where RMS is the mars -sun distance and RES is the earth-sun distance.